For any \(u(\cdot ) \in \mathcal {U}[0,T]\) , under Assumption REF ,
there exists a unique strong solution \(X^{u}(\cdot )\)  to the SDE in (REF ) ([1]}, Theorem 1.2; [2]}, Chapter I, Theorem 6.3) which satisfies the following standard estimate:
\(\mathbb {E}\left[ \sup \limits _{t\in [0,T]}\left|X^{u}(t)\right|^{8}\right] \le C\left\lbrace  \left|x_{0}\right|^{8}+\mathbb {E}\left[\left( \int _{0}^{T}\left|b(t,0,u(t))\right|dt\right) ^{8}+\left(\int _{0}^{T}\left|\sigma (t,0,u(t))\right|^{2}dt\right) ^{4}\right]\right\rbrace  .\) 
